Some theorems on the sum of nonlinear mappings of monotone type in Banach spaces (Q796088)

Let X be a reflexive Banach space with dual \(X^*\). In this paper equations of the form \(Au+Tu\ni f\) are studied, where \(f\in X^*\) is given and \(A:D(A)\subset X\to 2^{X^*}, T:D\subset(T)X\to 2^ X\) are nonlinear. It is further assumed that A is maximal monotone and has closed graph in \(X\times X^*.\) Results are obtained which indicate when solvability of the approximation equation \(A_{\lambda}u_{\lambda}+Tu_{\lambda}\ni f,\lambda>0,\) with \(A_{\lambda}\) the Yosida-approximation of A, imply the solvability of the equation \(Au+Tu\ni f\) as \(\lambda \to 0+.\) These results are then applied to the study of surjectivity of \(A+T:X\to 2^{X^*}.\) The initial results extend similar ones obtained by \textit{F. E. Browder} and \textit{P. Hess} [J. Funct. Anal. 11, 251-294 (1972; Zbl 0249.47044)], and the surjectivity results simplify and partially extend earlier ones of \textit{C. P. Gupta} [Proc. Am. Math. Soc. 53, 143-148 (1975; Zbl 0315.47035)].